Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for a particular derivation to ''correspond'' to a particular proof. Mere existence of a formalization is not enough, and a substantive account of the required correspondence resolves into two criteria -- adequate representation (of the original theorem) and tracking (of the steps in the original proof). An examination of the actually-existing formalization systems we have today shows the variety of quasi-empirical ways we establish these criteria, and points towards new burdens that may be placed on the future evolution of mathematics itself.
Alexey Sulavko, Irina Panfilova, Daniil Inivatov
et al.
This work is devoted to the development of a biometric authentication system and the generation of a cryptographic key or a long password of 1024 bits based on a voice password, which ensures the protection of a biometric template from compromise. A new hybrid neural network model based on two types of trigonometric correlation neurons was proposed. The model is capable of recording correlation links between features and is resistant to data extraction attacks. The experiments were conducted on our own AIC-spkr-130 dataset and the publicly available RedDots, including recordings of user voices in different psycho-emotional states (sleepy state, alcohol intoxication). The results show that the proposed neural fuzzy extractor model provides an equal error probability level of EER = 2.1%.
Jason T. Hedetniemi, Stephen T. Hedetniemi, Thomas M. Lewis
Let \(G = (V, E)\) be a graph with vertex set \(V\) and edge set \(E\). A set \(S \subset V\) is a \(2\)-packing in \(G\) if for any two vertices \(u,v \in S\), the distance between them satisfies \(d(u,v) \gt 2\). The upper \(2\)-packing number \(P_2(G)\) is the maximum cardinality of a \(2\)-packing in \(G\). A set \(S \subset V\) is a dominating set for \(G\) if every vertex in \(V - S\) is adjacent to at least one vertex in \(S\). The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set in \(G\). A set \(S \subset V\) is a distance-\(2\) dominating set if for every vertex \(v \in V - S\) there exists a vertex \(u \in S\) such that \(d(u,v) \leq 2\). The upper distance-\(2\) domination number \(\Gamma_{\leq 2}(G)\) is the maximum cardinality of a minimal distance-\(2\) dominating set in \(G\). In this paper we establish two families of graphs \(G\) for which \(P_2(G) = \gamma(G) = \Gamma_{\leq 2}(G)\), which extend several well-known equalities of the form \(P_2(G) = \gamma(G)\).
Henry Kwasi Asiedu, Benedict Barnes, Isaac Kwame Dontwi
et al.
A system of nonlinear fractional partial differential equations (FPDEs) is widely used in applied sciences, especially for modeling fluid dynamics and polymer-related problems. Given their importance, finding solutions to these systems is essential and a core property. Various methods have been developed to find a solution to a system of nonlinear FPDEs. However, these methods are difficult to implement and sometimes converge slowly. In the worst-case scenario, applying the differential transform method may produce a series that does not converge to the exact solution of a system of nonlinear FPDEs. The semi-separation of variables method (S-SVM) is a recent and reliable analytic method that has not been applied to obtain an exact solution to a system of nonlinear FPDEs. Furthermore, S-SVM has not been improved to observe faster convergence. In this paper, the S-SVM is used to obtain the exact solution to the system of nonlinear FPDEs. In addition, the S-SVM is further improved as a Modified S-SVM (MS-SVM), which is applied to find an exact solution to the system of nonlinear FPDEs. Also, numerical experiments using the S-SVM and the MS-SVM in both two and three dimensions are provided therein, along with a comparison of their solutions to those obtained from the Adomian Decomposition Method (ADM), the Laplace Variational Iteration Method (LVIM), and the Fractional Power Series Method (FPSM). The results show that the solutions obtained using S-SVM and MS-SVM converge faster than those from FPSM, ADM, and LVIM. Moreover, S-SVM and MS-SVM do not require the complex computation of Adomian polynomials.
Maria Kleshnina, Matthew Holden, Ryan Heneghan
et al.
There is a growing need to integrate sustainability into tertiary mathematics education given the urgency of addressing global environmental challenges. This paper presents four case studies from Australian university courses that incorporate ecological and environmentally-conscious concepts into the mathematics curriculum. These case studies cover topics such as population dynamics, sustainable fisheries management, statistical inference for endangered species assessment, and mathematical modelling of climate effects on marine ecosystems. Each case demonstrates how fundamental mathematical methods, including calculus, statistics and operations research, can be applied to real-world ecological issues. These examples are ready-to-implement problems for integrating ecological thinking into mathematics classes, providing educators with practical tools to help students develop interdisciplinary problem-solving skills and prepare for the challenges of sustainability in their future careers.
This report is the first of two publications of a joint Working Group of the International Mathematical Union (IMU) and the International Council of Industrial and Applied Mathematics (ICIAM). In it, we shall analyze the current state of publishing in the mathematical sciences and explain the resulting problems. Our second publication will offer concrete recommendations, guidelines, and best practices for researchers, policymakers, and evaluators of mathematical research. It will explain how to detect and counteract attempts to game bibliometric measures, empowering the community to reclaim control over research evaluation and drive necessary change.
In this paper, using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations, thereby establishing their complete integrability. The Gardner equation is chosen as the key example, as it comprises both the Korteweg–de Vries and modified Korteweg–de Vries equations. The Gardner and Miura transformations, which connect these equations, are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota’s method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case), whereas the focusing Gardner equation has standard elastically colliding solitons. This paper’s aim is to provide a review of the integrability properties and solutions of the Gardner equation and to illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in <i>Mathematica</i>, but can be adapted for major computer algebra systems.
Shakuntla Singla, Harshleen Kaur, Deepak Gupta
et al.
Scheduling is one of the many skills required for advancement in today’s modern industry. The flow-shop scheduling problem is a well-known combinatorial optimization challenge. Scheduling issues for flow shops are NP-hard and challenging. The present research investigates a two-stage flow shop scheduling problem with decoupled processing and setup times, where a correlation exists between probabilities, job processing times, and setup times. This study proposes a novel heuristic algorithm that optimally sequences jobs to minimize the makespan and eliminates machine idle time, thereby reducing machine rental costs. The proposed algorithm’s efficacy is demonstrated through several computational examples implemented in MATLAB 2021a. The results are compared with the existing approaches such as those by Johnson, Palmer, NEH, and Nailwal to highlight the proposed algorithm’s superior performance.
Himani Agarwal, Manvendra Narayan Mishra, Ravi Shanker Dubey
In this paper, we have attempted a fresh method to demonstrate how special functions and fractional calculus are used in real-world problems. Here, we have examined the glucose supply in human blood using the incomplete Aleph function (IAF) and the Caputo fractional operator. In this study, we used the incomplete Aleph function to find the blood glucose equation that is supplied to human blood. In terms of various hyper-geometric functions, we have also obtained several significant and unique results, and we have defined the blood glucose function in terms of IAF.
It has been reported by World Health Organization (WHO) that the Covid-19 epidemic due to the Sar Cov-2 virus, which started in China and affected the whole world, caused the death of approximately six million people over three years. Global disasters such as pandemics not only cause deaths but also bring other global catastrophic problems. Therefore, governments need to perform very serious strategic operations to prevent both infection and death. It is accepted that even if there are vaccines developed against the virus, it will never be possible to predict very complex spread dynamics and reach a spread pattern due to new variants and other parameters. In the present study, four countries: Türkiye, Germany, Italy, and the United Kingdom have been selected since they exhibit similar characteristics in terms of the pandemic’s onset date, wave patterns, measures taken against the outbreak, and the vaccines used. Additionally, they are all located on the same continent. For these reasons, the three-year Covid-19 data of these countries were analyzed. Detailed chaotic attractors analyses were performed for each country and Lyapunov exponents were obtained. We showed that the three-year times series is chaotic for the chosen countries. In this sense, our results are compatible with the results of the Covid-19 analysis results in the literature. However, unlike previous Covid-19 studies, we also found out that there are chaotic, periodic, or quasi-periodic sub-series within these chaotic time series. The obtained results are of great importance in terms of revealing the details of the dynamics of the pandemic.
Mathematical modeling plays a significant role in understanding and controlling Ebola outbreaks. This study focuses on investigating analytical approximate solutions to the nonlinear Ebola epidemic Susceptible-Exposed-Infectious-Recovered (SEIR) model. The perturbation technique, namely the homotopy perturbation method (HPM) is utilized in this study. The SEIR model is crucial for sketching the dynamics of Ebola transmission, including the progression of individuals through different disease stages. To validate the approximate results derived from the HPM, we compare them with solutions obtained using the Runge-Kutta fourth-order (RK4) method. The comparison reveals excellent agreement between the HPM and RK4 solutions, confirming the accuracy of the analytic approach and the effectiveness of the HPM in solving complex epidemic models. Furthermore, graphical illustrations of the results provide valuable insights into the behavior and progression of Ebola outbreaks over time. These illustrations highlight the potential of the HPM as a powerful tool for investigating epidemic models and the development of control strategies.
The phase field model is a widely used mathematical approach for describing crack propagation in continuum damage fractures. In the context of phase field fracture simulations, adaptive finite element methods (AFEM) are often employed to address the mesh size dependency of the model. However, existing AFEM approaches for this application frequently rely on heuristic adjustments and empirical parameters for mesh refinement. In this paper, we introduce an adaptive finite element method based on a recovery type posteriori error estimates approach grounded in theoretical analysis. This method transforms the gradient of the numerical solution into a smoother function space, using the difference between the recovered gradient and the original numerical gradient as an error indicator for adaptive mesh refinement. This enables the automatic capture of crack propagation directions without the need for empirical parameters. We have implemented this adaptive method for the Hybrid formulation of the phase field model using the open-source software package FEALPy. The accuracy and efficiency of the proposed approach are demonstrated through simulations of classical 2D and 3D brittle fracture examples, validating the robustness and effectiveness of our implementation.
Abstract The purpose of this paper is to study the convergence theorems in CAT ( κ ) $\operatorname{CAT} (\kappa )$ spaces with k > 0 $k > 0$ for total asymptotically nonexpansive mappings which are essentially wider than nonexpansive mappings, asymptotically nonexpansive mapping, and asymptotically nonexpansive mappings in the intermediate sense. Our results generalize, unify, and improve several comparable results in the existing literature.
In this paper, we propose a novel methodology that combines the opposition Nelder–Mead algorithm and the selection phase of the genetic algorithm. This integration aims to enhance the performance of the overall algorithm. To evaluate the effectiveness of our methodology, we conducted a comprehensive comparative study involving 11 state-of-the-art algorithms renowned for their exceptional performance in the 2022 IEEE Congress on Evolutionary Computation (CEC 2022). Following rigorous analysis, which included a Friedman test and subsequent Dunn’s post hoc test, our algorithm demonstrated outstanding performance. In fact, our methodology exhibited equal or superior performance compared to the other algorithms in the majority of cases examined. These results highlight the effectiveness and competitiveness of our proposed approach, showcasing its potential to achieve state-of-the-art performance in solving optimization problems.
Katsunobu Sasanuma, Robert Hampshire, Alan Scheller-Wolf
The Erlang A model – an M/M/s queue with exponential abandonment – is often used to represent a service system with impatient customers. For this system, the popular square-root staffing rule determines the necessary staffing level to achieve the desirable QED (quality-and-efficiency-driven) service regime; however, the rule also implies that properties of large systems are highly sensitive to parameters. We reveal that the origin of this high sensitivity is due to the operation of large systems at a point of singularity in a phase diagram of service regimes. We can avoid this singularity by implementing a congestion-based control (CBC) scheme—a scheme that allows the system to change its arrival and service rates under congestion. We analyze a modified Erlang A model under the CBC scheme using a Markov chain decomposition method, derive non-asymptotic and asymptotic normal representations of performance indicators, and confirm that the CBC scheme makes large systems less sensitive than the original Erlang A model.